Definition. An ellipse is the locus of points in a plane, the sum of the distances of each of them from two given points of this plane, called foci, is a constant value (provided that this value is greater than the distance between the foci).
Let's denote the foci through the distance between them - through , and a constant value equal to the sum of the distances from each point of the ellipse to the foci, through (by condition ).
Let's build a Cartesian coordinate system so that the foci are on the abscissa axis, and the origin of coordinates coincides with the middle of the segment (Fig. 44). Then the focuses will have the following coordinates: left focus and right focus. Let's derive the equation of the ellipse in the coordinate system we have chosen. To this end, consider an arbitrary point of the ellipse. By definition of an ellipse, the sum of the distances from this point to the foci is:
Using the formula for the distance between two points, we obtain, therefore,
To simplify this equation, we write it in the form
Then squaring both sides of the equation gives
or, after obvious simplifications:
Now again we square both sides of the equation, after which we will have:
or, after identical transformations:
Since according to the condition in the definition of an ellipse , then is a positive number. We introduce the notation
Then the equation will take the following form:
By definition of an ellipse, the coordinates of any of its points satisfy equation (26). But equation (29) is a consequence of equation (26). Therefore, it also satisfies the coordinates of any point of the ellipse.
It can be shown that the coordinates of points that do not lie on the ellipse do not satisfy equation (29). Thus, equation (29) is the equation of an ellipse. It is called the canonical equation of the ellipse.
Let's establish the shape of the ellipse using its canonical equation.
First of all, note that this equation contains only even powers of x and y. This means that if any point belongs to an ellipse, then it also includes a point that is symmetrical with a point about the abscissa axis, and a point that is symmetric with a point about the y-axis. Thus, the ellipse has two mutually perpendicular axes of symmetry, which in our chosen coordinate system coincide with the coordinate axes. The axes of symmetry of the ellipse will be called the axes of the ellipse, and the point of their intersection - the center of the ellipse. The axis on which the foci of the ellipse are located (in this case, the abscissa axis) is called the focal axis.
Let's determine the shape of the ellipse first in the first quarter. To do this, we solve equation (28) with respect to y:
It is obvious that here , since y takes imaginary values for . With an increase from 0 to a, y decreases from b to 0. The part of the ellipse lying in the first quarter will be an arc bounded by points B (0; b) and lying on the coordinate axes (Fig. 45). Using now the symmetry of the ellipse, we conclude that the ellipse has the shape shown in Fig. 45.
The points of intersection of the ellipse with the axes are called the vertices of the ellipse. It follows from the symmetry of the ellipse that, in addition to the vertices, the ellipse has two more vertices (see Fig. 45).
The segments and connecting the opposite vertices of the ellipse, as well as their lengths, are called the major and minor axes of the ellipse, respectively. The numbers a and b are called the major and minor semiaxes of the ellipse, respectively.
The ratio of half the distance between the foci to the semi-major axis of the ellipse is called the eccentricity of the ellipse and is usually denoted by the letter:
Since , then the eccentricity of the ellipse is less than one: The eccentricity characterizes the shape of the ellipse. Indeed, it follows from formula (28), From this it can be seen that the smaller the eccentricity of the ellipse, the less its minor semiaxis b differs from the major semiaxis a, i.e., the less the ellipse is extended (along the focal axis).
In the limiting case, when you get a circle of radius a: , or . At the same time, the foci of the ellipse, as it were, merge at one point - the center of the circle. The eccentricity of the circle is zero:
The connection between the ellipse and the circle can be established from another point of view. Let us show that an ellipse with semi-axes a and b can be considered as a projection of a circle of radius a.
Let us consider two planes P and Q, forming such an angle a between themselves, for which (Fig. 46). Let us construct a coordinate system in the P plane, and an Oxy system in the Q plane with a common origin O and a common abscissa axis coinciding with the line of intersection of the planes. Consider in the plane P the circle
centered at the origin and radius a. Let be an arbitrarily chosen point of the circle, be its projection onto the Q plane, and be the projection of the point M onto the Ox axis. Let us show that the point lies on an ellipse with semi-axes a and b.
Lines of the second order.
Ellipse and its canonical equation. Circle
After a thorough study straight lines on the plane we continue to study the geometry of the two-dimensional world. The stakes are doubled and I invite you to visit the picturesque gallery of ellipses, hyperbolas, parabolas, which are typical representatives of second order lines. The tour has already begun, and first, a brief information about the entire exhibition on different floors of the museum:
The concept of an algebraic line and its order
A line on a plane is called algebraic, if in affine coordinate system its equation has the form , where is a polynomial consisting of terms of the form ( is a real number, are non-negative integers).
As you can see, the equation of an algebraic line does not contain sines, cosines, logarithms and other functional beau monde. Only "x" and "y" in integer non-negative degrees.
Line order is equal to the maximum value of the terms included in it.
According to the corresponding theorem, the concept of an algebraic line, as well as its order, do not depend on the choice affine coordinate system, therefore, for the ease of being, we consider that all subsequent calculations take place in Cartesian coordinates.
General Equation the second-order line has the form , where 
are arbitrary real numbers (it is customary to write with a multiplier - "two"), and the coefficients are not simultaneously equal to zero.
If , then the equation simplifies to 
, and if the coefficients are not simultaneously equal to zero, then this is exactly general equation of a "flat" straight line, which represents first order line.
Many understood the meaning of the new terms, but, nevertheless, in order to 100% assimilate the material, we stick our fingers into the socket. To determine the line order, iterate over all terms its equations and for each of them find sum of powers incoming variables.
For example:
the term contains "x" to the 1st degree;
the term contains "Y" to the 1st power;
there are no variables in the term, so the sum of their powers is zero.
Now let's figure out why the equation sets the line second order:
the term contains "x" in the 2nd degree;
the term has the sum of the degrees of the variables: 1 + 1 = 2;
the term contains "y" in the 2nd degree;
all other terms - lesser degree.
Maximum value: 2
If we additionally add to our equation, say, , then it will already determine third order line. It is obvious that the general form of the 3rd order line equation contains a “complete set” of terms, the sum of the degrees of variables in which is equal to three:
, where the coefficients are not simultaneously equal to zero.
In the event that one or more suitable terms are added that contain 
, then we will talk about 4th order lines, etc.
We will have to deal with algebraic lines of the 3rd, 4th and higher orders more than once, in particular, when getting acquainted with polar coordinate system.
However, let us return to the general equation and recall its simplest school variations. Examples are the parabola, whose equation can be easily reduced to a general form, and the hyperbola with an equivalent equation. However, not everything is so smooth ....
A significant drawback of the general equation is that it is almost always not clear which line it defines. Even in the simplest case, you will not immediately realize that this is hyperbole. Such layouts are good only at a masquerade, therefore, in the course of analytical geometry, a typical problem is considered reduction of the 2nd order line equation to the canonical form.
What is the canonical form of an equation?
This is the generally accepted standard form of the equation, when in a matter of seconds it becomes clear what geometric object it defines. In addition, the canonical form is very convenient for solving many practical problems. So, for example, according to the canonical equation "flat" straight, firstly, it is immediately clear that this is a straight line, and secondly, the point belonging to it and the direction vector are simply visible.
Obviously, any 1st order line represents a straight line. On the second floor, there is no longer a janitor waiting for us, but a much more diverse company of nine statues:
Classification of second order lines
With the help of a special set of actions, any second-order line equation is reduced to one of the following types:
( and are positive real numbers)
1) 
is the canonical equation of the ellipse;
2) is the canonical equation of the hyperbola;
3) 
is the canonical equation of the parabola;
4) – imaginary ellipse;
5) - a pair of intersecting lines;
6) - couple imaginary intersecting lines (with the only real point of intersection at the origin);
7) - a pair of parallel lines;
8) - couple imaginary parallel lines;
9) is a pair of coinciding lines.
Some readers may get the impression that the list is incomplete. For example, in paragraph number 7, the equation sets the pair direct, parallel to the axis, and the question arises: where is the equation that determines the lines parallel to the ordinate axis? Answer: it not considered canon. The straight lines represent the same standard case rotated by 90 degrees, and an additional entry in the classification is redundant, since it does not carry anything fundamentally new.
Thus, there are nine and only nine different types of 2nd order lines, but in practice the most common are ellipse, hyperbola and parabola.
Let's look at the ellipse first. As usual, I focus on those points that are of great importance for solving problems, and if you need a detailed derivation of formulas, proofs of theorems, please refer, for example, to the textbook by Bazylev / Atanasyan or Aleksandrov.
Ellipse and its canonical equation
Spelling ... please do not repeat the mistakes of some Yandex users who are interested in "how to build an ellipse", "the difference between an ellipse and an oval" and "elebs eccentricity".
The canonical equation of an ellipse has the form , where are positive real numbers, and . I will formulate the definition of an ellipse later, but for now it's time to take a break from talking and solve a common problem:
How to build an ellipse?
Yes, take it and just draw it. The assignment is common, and a significant part of the students do not quite competently cope with the drawing:
Example 1
Construct an ellipse given by the equation
Decision: first we bring the equation to the canonical form: 
Why bring? One of the advantages of the canonical equation is that it allows you to instantly determine ellipse vertices, which are at the points . It is easy to see that the coordinates of each of these points satisfy the equation .
In this case : 
Line segment called major axis ellipse;
line segment – minor axis;
number 
called semi-major axis ellipse;
number 
– semi-minor axis.
in our example: .
To quickly imagine what this or that ellipse looks like, just look at the values \u200b\u200bof "a" and "be" of its canonical equation.
Everything is fine, neat and beautiful, but there is one caveat: I completed the drawing using the program. And you can draw with any application. However, in harsh reality, a checkered piece of paper lies on the table, and mice dance around our hands. People with artistic talent, of course, can argue, but you also have mice (albeit smaller ones). It is not in vain that mankind invented a ruler, a compass, a protractor and other simple devices for drawing.
For this reason, we are unlikely to be able to accurately draw an ellipse, knowing only the vertices. Still all right, if the ellipse is small, for example, with semiaxes. Alternatively, you can reduce the scale and, accordingly, the dimensions of the drawing. But in the general case it is highly desirable to find additional points.
There are two approaches to constructing an ellipse - geometric and algebraic. I don’t like building with a compass and ruler because of the short algorithm and the significant clutter of the drawing. In case of emergency, please refer to the textbook, but in reality it is much more rational to use the tools of algebra. From the ellipse equation on the draft, we quickly express: 
The equation is then split into two functions: 
– defines the upper arc of the ellipse; 
– defines the lower arc of the ellipse.
The ellipse given by the canonical equation is symmetrical with respect to the coordinate axes, as well as with respect to the origin. And that's great - symmetry is almost always a harbinger of a freebie. Obviously, it is enough to deal with the 1st coordinate quarter, so we need a function 
. It suggests finding additional points with abscissas 
. We hit three SMS on the calculator: 
Of course, it is also pleasant that if a serious error is made in the calculations, then this will immediately become clear during the construction.
Mark points on the drawing (red color), symmetrical points on the other arcs (blue color) and carefully connect the whole company with a line: 
It is better to draw the initial sketch thinly and thinly, and only then apply pressure to the pencil. The result should be quite a decent ellipse. By the way, would you like to know what this curve is?
Definition of an ellipse. Ellipse foci and ellipse eccentricity
An ellipse is a special case of an oval. The word "oval" should not be understood in the philistine sense ("the child drew an oval", etc.). This is a mathematical term with a detailed formulation. The purpose of this lesson is not to consider the theory of ovals and their various types, which are practically not given attention in the standard course of analytic geometry. And, in accordance with more current needs, we immediately go to the strict definition of an ellipse:
Ellipse- this is the set of all points of the plane, the sum of the distances to each of which from two given points, called tricks ellipse, is a constant value, numerically equal to the length of the major axis of this ellipse: .
In this case, the distance between the foci is less than this value: .
Now it will become clearer: 
Imagine that the blue dot "rides" on an ellipse. So, no matter what point of the ellipse we take, the sum of the lengths of the segments will always be the same:
Let's make sure that in our example the value of the sum is really equal to eight. Mentally place the point "em" in the right vertex of the ellipse, then: , which was required to be checked.
Another way to draw an ellipse is based on the definition of an ellipse. Higher mathematics, at times, is the cause of tension and stress, so it's time to have another session of unloading. Please take a piece of paper or a large sheet of cardboard and pin it to the table with two nails. These will be tricks. Tie a green thread to the protruding nail heads and pull it all the way with a pencil. The neck of the pencil will be at some point, which belongs to the ellipse. Now begin to guide the pencil across the sheet of paper, keeping the green thread very taut. Continue the process until you return to the starting point ... excellent ... the drawing can be submitted for verification by the doctor to the teacher =)
How to find the focus of an ellipse?
In the above example, I depicted "ready" focus points, and now we will learn how to extract them from the depths of geometry.
If the ellipse is given by the canonical equation , then its foci have coordinates 
, where is it distance from each of the foci to the center of symmetry of the ellipse.
Calculations are easier than steamed turnips: 
! With the meaning "ce" it is impossible to identify the specific coordinates of tricks! I repeat, this is DISTANCE from each focus to the center(which in the general case does not have to be located exactly at the origin).
And, therefore, the distance between the foci cannot be tied to the canonical position of the ellipse either. In other words, the ellipse can be moved to another place and the value will remain unchanged, while the foci will naturally change their coordinates. Please consider this moment during further study of the topic.
The eccentricity of an ellipse and its geometric meaning
The eccentricity of an ellipse is a ratio that can take values within .
In our case:
Let's find out how the shape of an ellipse depends on its eccentricity. For this fix the left and right vertices of the ellipse under consideration, that is, the value of the semi-major axis will remain constant. Then the eccentricity formula will take the form: .
Let's start to approximate the value of the eccentricity to unity. This is only possible if . What does it mean? ...remembering tricks 
. This means that the foci of the ellipse will "disperse" along the abscissa axis to the side vertices. And, since “the green segments are not rubber”, the ellipse will inevitably begin to flatten, turning into a thinner and thinner sausage strung on an axis.
Thus, the closer the eccentricity of the ellipse is to one, the more oblong the ellipse is.
Now let's simulate the opposite process: the foci of the ellipse 
went towards each other, approaching the center. This means that the value of "ce" is getting smaller and, accordingly, the eccentricity tends to zero: .
In this case, the “green segments”, on the contrary, will “become crowded” and they will begin to “push” the line of the ellipse up and down.
Thus, the closer the eccentricity value is to zero, the more the ellipse looks like... look at the limiting case, when the foci are successfully reunited at the origin:
A circle is a special case of an ellipse
Indeed, in the case of equality of the semiaxes, the canonical equation of the ellipse takes the form, which reflexively transforms to the well-known circle equation from the school with the center at the origin of the radius "a".
In practice, the notation with the “speaking” letter “er” is more often used:. The radius is called the length of the segment, while each point of the circle is removed from the center by the distance of the radius.
Note that the definition of an ellipse remains completely correct: the foci matched, and the sum of the lengths of the matched segments for each point on the circle is a constant value. Since the distance between foci is the eccentricity of any circle is zero.
A circle is built easily and quickly, it is enough to arm yourself with a compass. However, sometimes it is necessary to find out the coordinates of some of its points, in this case we go the familiar way - we bring the equation to a cheerful Matan's form:
is the function of the upper semicircle;
is the function of the lower semicircle.
Then we find the desired values, differentiable, integrate and do other good things.
The article, of course, is for reference only, but how can one live without love in the world? Creative task for independent solution
Example 2
Compose the canonical equation of an ellipse if one of its foci and the semi-minor axis are known (the center is at the origin). Find vertices, additional points and draw a line on the drawing. Calculate the eccentricity.
Solution and drawing at the end of the lesson
Let's add an action:
Rotate and translate an ellipse
Let's return to the canonical equation of the ellipse, namely, to the condition, the riddle of which has been tormenting inquisitive minds since the first mention of this curve. Here we have considered an ellipse 
, but in practice can not the equation 
? After all, here, however, it seems to be like an ellipse too!
Such an equation is rare, but it does come across. And it does define an ellipse. Let's dispel the mystic: 
As a result of the construction, our native ellipse is obtained, rotated by 90 degrees. I.e, 
- This non-canonical entry ellipse 
. Record!- the equation 
does not specify any other ellipse, since there are no points (foci) on the axis that would satisfy the definition of an ellipse.
An ellipse is the locus of points in a plane, the sum of the distances from each of them to two given points F_1, and F_2 is a constant value (2a), greater than the distance (2c) between these given points (Fig. 3.36, a). This geometric definition expresses focal property of an ellipse.
Focal property of an ellipse
Points F_1 and F_2 are called the foci of the ellipse, the distance between them 2c=F_1F_2 is the focal length, the midpoint O of the segment F_1F_2 is the center of the ellipse, the number 2a is the length of the major axis of the ellipse (respectively, the number a is the major semiaxis of the ellipse). The segments F_1M and F_2M connecting an arbitrary point M of the ellipse with its foci are called the focal radii of the point M . A line segment connecting two points of an ellipse is called a chord of the ellipse.
The ratio e=\frac(c)(a) is called the eccentricity of the ellipse. From the definition (2a>2c) it follows that 0\leqslant e<1 . При e=0 , т.е. при c=0 , фокусы F_1 и F_2 , а также центр O совпадают, и эллипс является окружностью радиуса a (рис.3.36,6).
Geometric definition of an ellipse, expressing its focal property, is equivalent to its analytical definition - a line given by the canonical equation of an ellipse:
Indeed, let's introduce a rectangular coordinate system (Fig. 3.36, c). The center O of the ellipse is taken as the origin of the coordinate system; the straight line passing through the foci (the focal axis or the first axis of the ellipse), we will take as the abscissa axis (the positive direction on it from the point F_1 to the point F_2); the straight line perpendicular to the focal axis and passing through the center of the ellipse (the second axis of the ellipse) is taken as the y-axis (the direction on the y-axis is chosen so that the rectangular coordinate system Oxy is right).
Let us formulate the equation of an ellipse using its geometric definition, which expresses the focal property. In the selected coordinate system, we determine the coordinates of the foci F_1(-c,0),~F_2(c,0). For an arbitrary point M(x,y) belonging to the ellipse, we have:
\vline\,\overrightarrow(F_1M)\,\vline\,+\vline\,\overrightarrow(F_2M)\,\vline\,=2a.
Writing this equality in coordinate form, we get:
\sqrt((x+c)^2+y^2)+\sqrt((x-c)^2+y^2)=2a.
We transfer the second radical to the right side, square both sides of the equation and give like terms:
(x+c)^2+y^2=4a^2-4a\sqrt((x-c)^2+y^2)+(x-c)^2+y^2~\Leftrightarrow ~4a\sqrt((x-c )^2+y^2)=4a^2-4cx.
Dividing by 4, we square both sides of the equation:
A^2(x-c)^2+a^2y^2=a^4-2a^2cx+c^2x^2~\Leftrightarrow~ (a^2-c^2)^2x^2+a^2y^ 2=a^2(a^2-c^2).
Denoting b=\sqrt(a^2-c^2)>0, we get b^2x^2+a^2y^2=a^2b^2. Dividing both parts by a^2b^2\ne0 , we arrive at the canonical equation of the ellipse:
\frac(x^2)(a^2)+\frac(y^2)(b^2)=1.
Therefore, the chosen coordinate system is canonical.
If the foci of the ellipse coincide, then the ellipse is a circle (Fig. 3.36.6), since a=b. In this case, any rectangular coordinate system with origin at the point O\equiv F_1\equiv F_2, and the equation x^2+y^2=a^2 is the equation of a circle with center O and radius a .
By reasoning backwards, it can be shown that all the points whose coordinates satisfy equation (3.49), and only they, belong to the locus of points, called the ellipse. In other words, the analytic definition of an ellipse is equivalent to its geometric definition, which expresses the focal property of the ellipse.
Directory property of an ellipse
The directrixes of an ellipse are two straight lines passing parallel to the ordinate axis of the canonical coordinate system at the same distance \frac(a^2)(c) from it. For c=0 , when the ellipse is a circle, there are no directrixes (we can assume that the directrixes are infinitely removed).
Ellipse with eccentricity 0
Indeed, for example, for focus F_2 and directrix d_2 (Fig. 3.37.6) the condition \frac(r_2)(\rho_2)=e can be written in coordinate form:
\sqrt((x-c)^2+y^2)=e\cdot\!\left(\frac(a^2)(c)-x\right)
Getting rid of irrationality and replacing e=\frac(c)(a),~a^2-c^2=b^2, we arrive at the canonical equation of the ellipse (3.49). Similar reasoning can be carried out for the focus F_1 and the directrix d_1\colon\frac(r_1)(\rho_1)=e.
Ellipse equation in polar coordinates
The ellipse equation in the polar coordinate system F_1r\varphi (Fig.3.37,c and 3.37(2)) has the form
R=\frac(p)(1-e\cdot\cos\varphi)
where p=\frac(b^2)(a) is the focal parameter of the ellipse.
In fact, let's choose the left focus F_1 of the ellipse as the pole of the polar coordinate system, and the ray F_1F_2 as the polar axis (Fig. 3.37, c). Then for an arbitrary point M(r,\varphi) , according to the geometric definition (focal property) of an ellipse, we have r+MF_2=2a . We express the distance between the points M(r,\varphi) and F_2(2c,0) (see point 2 of remarks 2.8):
\begin(aligned)F_2M&=\sqrt((2c)^2+r^2-2\cdot(2c)\cdot r\cos(\varphi-0))=\\ &=\sqrt(r^2- 4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2).\end(aligned)
Therefore, in coordinate form, the equation of the ellipse F_1M+F_2M=2a has the form
R+\sqrt(r^2-4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2)=2\cdot a.
We isolate the radical, square both sides of the equation, divide by 4 and give like terms:
R^2-4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2~\Leftrightarrow~a\cdot\!\left(1-\frac(c)(a)\cdot\cos \varphi\right)\!\cdot r=a^2-c^2.
We express the polar radius r and make the substitution e=\frac(c)(a),~b^2=a^2-c^2,~p=\frac(b^2)(a):
R=\frac(a^2-c^2)(a\cdot(1-e\cdot\cos\varphi)) \quad \Leftrightarrow \quad r=\frac(b^2)(a\cdot(1 -e\cdot\cos\varphi)) \quad \Leftrightarrow \quad r=\frac(p)(1-e\cdot\cos\varphi),
Q.E.D.
The geometric meaning of the coefficients in the ellipse equation
Let's find the intersection points of the ellipse (see Fig. 3.37, a) with the coordinate axes (vertices of the zllips). Substituting y=0 into the equation, we find the intersection points of the ellipse with the abscissa axis (with the focal axis): x=\pm a . Therefore, the length of the segment of the focal axis enclosed within the ellipse is equal to 2a. This segment, as noted above, is called the major axis of the ellipse, and the number a is the major semi-axis of the ellipse. Substituting x=0 , we get y=\pm b . Therefore, the length of the segment of the second axis of the ellipse enclosed inside the ellipse is equal to 2b. This segment is called the minor axis of the ellipse, and the number b is called the minor semiaxis of the ellipse.
Really, b=\sqrt(a^2-c^2)\leqslant\sqrt(a^2)=a, and the equality b=a is obtained only in the case c=0 when the ellipse is a circle. Attitude k=\frac(b)(a)\leqslant1 is called the contraction factor of the ellipse.
Remarks 3.9
1. The lines x=\pm a,~y=\pm b limit the main rectangle on the coordinate plane, inside which the ellipse is located (see Fig. 3.37, a).
2. An ellipse can be defined as the locus of points obtained by contracting a circle to its diameter.
Indeed, let in the rectangular coordinate system Oxy the circle equation has the form x^2+y^2=a^2 . When compressed to the x-axis with a factor of 0 \begin(cases)x"=x,\\y"=k\cdot y.\end(cases) Substituting x=x" and y=\frac(1)(k)y" into the equation of the circle, we obtain an equation for the coordinates of the image M"(x",y") of the point M(x,y) : (x")^2+(\left(\frac(1)(k)\cdot y"\right)\^2=a^2 \quad \Leftrightarrow \quad \frac{(x")^2}{a^2}+\frac{(y")^2}{k^2\cdot a^2}=1 \quad \Leftrightarrow \quad \frac{(x")^2}{a^2}+\frac{(y")^2}{b^2}=1,
!} since b=k\cdot a . This is the canonical equation of the ellipse. 3. The coordinate axes (of the canonical coordinate system) are the axes of symmetry of the ellipse (called the principal axes of the ellipse), and its center is the center of symmetry. Indeed, if the point M(x,y) belongs to the ellipse . then the points M"(x,-y) and M""(-x,y) , symmetrical to the point M with respect to the coordinate axes, also belong to the same ellipse. 4. From the equation of an ellipse in a polar coordinate system r=\frac(p)(1-e\cos\varphi)(see Fig. 3.37, c), the geometric meaning of the focal parameter is clarified - this is half the length of the chord of the ellipse passing through its focus perpendicular to the focal axis ( r = p at \varphi=\frac(\pi)(2)). 5. The eccentricity e characterizes the shape of the ellipse, namely the difference between the ellipse and the circle. The larger e, the more elongated the ellipse, and the closer e is to zero, the closer the ellipse is to the circle (Fig. 3.38, a). Indeed, given that e=\frac(c)(a) and c^2=a^2-b^2 , we get E^2=\frac(c^2)(a^2)=\frac(a^2-b^2)(a^2)=1-(\left(\frac(a)(b)\right )\^2=1-k^2,
!} where k is the contraction factor of the ellipse, 0 6. Equation \frac(x^2)(a^2)+\frac(y^2)(b^2)=1 for a
7. Equation \frac((x-x_0)^2)(a^2)+\frac((y-y_0)^2)(b^2)=1,~a\geqslant b defines an ellipse centered at the point O "(x_0, y_0), whose axes are parallel to the coordinate axes (Fig. 3.38, c). This equation is reduced to the canonical one using parallel translation (3.36). For a=b=R the equation (x-x_0)^2+(y-y_0)^2=R^2 describes a circle of radius R centered at point O"(x_0,y_0) . Parametric equation of an ellipse in the canonical coordinate system has the form \begin(cases)x=a\cdot\cos(t),\\ y=b\cdot\sin(t),\end(cases)0\leqslant t<2\pi.
Indeed, substituting these expressions into equation (3.49), we arrive at the basic trigonometric identity \cos^2t+\sin^2t=1 . Example 3.20. draw ellipse \frac(x^2)(2^2)+\frac(y^2)(1^2)=1 in the canonical coordinate system Oxy . Find semiaxes, focal length, eccentricity, aspect ratio, focal parameter, directrix equations. Decision. Comparing the given equation with the canonical one, we determine the semiaxes: a=2 - the major semiaxis, b=1 - the minor semiaxis of the ellipse. We build the main rectangle with sides 2a=4,~2b=2 centered at the origin (Fig.3.39). Given the symmetry of the ellipse, we fit it into the main rectangle. If necessary, we determine the coordinates of some points of the ellipse. For example, substituting x=1 into the ellipse equation, we get \frac(1^2)(2^2)+\frac(y^2)(1^2)=1 \quad \Leftrightarrow \quad y^2=\frac(3)(4) \quad \Leftrightarrow \ quad y=\pm\frac(\sqrt(3))(2). Therefore, points with coordinates \left(1;\,\frac(\sqrt(3))(2)\right)\!,~\left(1;\,-\frac(\sqrt(3))(2)\right)- belong to an ellipse. Calculate the compression ratio k=\frac(b)(a)=\frac(1)(2); focal length 2c=2\sqrt(a^2-b^2)=2\sqrt(2^2-1^2)=2\sqrt(3); eccentricity e=\frac(c)(a)=\frac(\sqrt(3))(2); focal parameter p=\frac(b^2)(a)=\frac(1^2)(2)=\frac(1)(2). We compose the directrix equations: x=\pm\frac(a^2)(c)~\Leftrightarrow~x=\pm\frac(4)(\sqrt(3)). 11.1. Basic concepts Consider the lines defined by equations of the second degree with respect to the current coordinates The coefficients of the equation are real numbers, but at least one of the numbers A, B, or C is non-zero. Such lines are called lines (curves) of the second order. It will be established below that equation (11.1) defines a circle, ellipse, hyperbola, or parabola in the plane. Before proceeding to this assertion, let us study the properties of the enumerated curves. 11.2. Circle Then from the condition we obtain the equation Equation (11.2) is satisfied by the coordinates of any point on the given circle and is not satisfied by the coordinates of any point that does not lie on the circle. Equation (11.2) is called the canonical equation of the circle In particular, assuming and , we obtain the equation of a circle centered at the origin The circle equation (11.2) after simple transformations will take the form . When comparing this equation with the general equation (11.1) of a second-order curve, it is easy to see that two conditions are satisfied for the equation of a circle: 1) the coefficients at x 2 and y 2 are equal to each other; 2) there is no member containing the xy product of the current coordinates. Let's consider the inverse problem. Putting in equation (11.1) the values and , we obtain Let's transform this equation: It follows that equation (11.3) defines a circle under the condition If It is satisfied by the coordinates of a single point If a 11.3. Ellipse Canonical equation of an ellipse Ellipse
is the set of all points of the plane, the sum of the distances from each of them to two given points of this plane, called tricks
, is a constant value greater than the distance between the foci. To derive the equation of an ellipse, we choose a coordinate system so that the foci F1 and F2 lie on the axis , and the origin coincides with the midpoint of the segment F 1 F 2. Then the foci will have the following coordinates: and . Let be an arbitrary point of the ellipse. Then, according to the definition of an ellipse, i.e. This, in fact, is the equation of an ellipse. We transform equation (11.5) to a simpler form as follows: As a>with, then . Let's put Then the last equation takes the form or It can be proved that equation (11.7) is equivalent to the original equation. It's called the canonical equation of the ellipse
. Ellipse is a curve of the second order. Study of the shape of an ellipse according to its equation Let's establish the shape of the ellipse using its canonical equation. 1. Equation (11.7) contains x and y only in even powers, so if a point belongs to an ellipse, then points ,, also belong to it. It follows that the ellipse is symmetrical with respect to the axes and , as well as with respect to the point , which is called the center of the ellipse. 3. It follows from equation (11.7) that each term on the left-hand side does not exceed one, i.e. there are inequalities and or and . Therefore, all points of the ellipse lie inside the rectangle formed by the straight lines. 4. In equation (11.7), the sum of non-negative terms and is equal to one. Consequently, as one term increases, the other will decrease, that is, if it increases, then it decreases and vice versa. From what has been said, it follows that the ellipse has the shape shown in Fig. 50 (oval closed curve). More about the ellipse The shape of the ellipse depends on the ratio. When the ellipse turns into a circle, the ellipse equation (11.7) takes the form . As a characteristic of the shape of an ellipse, the ratio is more often used. The ratio of half the distance between the foci to the semi-major axis of the ellipse is called the eccentricity of the ellipse and o6o is denoted by the letter ε ("epsilon"): with 0<ε< 1, так как 0<с<а. С учетом равенства (11.6) формулу
(11.8) можно переписать в виде This shows that the smaller the eccentricity of the ellipse, the less oblate the ellipse will be; if we put ε = 0, then the ellipse turns into a circle. There are formulas Straight lines are called It follows from equality (11.6) that . If , then equation (11.7) defines an ellipse, the major axis of which lies on the Oy axis, and the minor axis lies on the Ox axis (see Fig. 52). The foci of such an ellipse are at the points and , where 11.4. Hyperbola Canonical equation of a hyperbola Hyperbole
is called the set of all points of the plane, the modulus of the difference in distances from each of which to two given points of this plane, called tricks
, is a constant value, smaller than the distance between the foci. To derive the hyperbola equation, we choose a coordinate system so that the foci F1 and F2 lie on the axis , and the origin coincided with the midpoint of the segment F 1 F 2(see fig. 53). Then the foci will have coordinates and Let be an arbitrary point of the hyperbola. Then according to the definition of a hyperbola A hyperbola is a line of the second order. Investigation of the form of a hyperbola according to its equation Let us establish the shape of the hyperbola using its caconic equation. 1. Equation (11.9) contains x and y only in even powers. Therefore, the hyperbola is symmetrical with respect to the axes and , as well as with respect to the point , which is called the center of the hyperbola.
2. Find the intersection points of the hyperbola with the coordinate axes. Putting in equation (11.9), we find two points of intersection of the hyperbola with the axis : and . Putting in (11.9), we obtain , which cannot be. Therefore, the hyperbola does not intersect the y-axis. The points and are called
peaks
hyperbolas, and the segment real axis
, line segment - real semiaxis
hyperbole. The line segment connecting the points is called
imaginary axis
, number b - imaginary axis
. Rectangle with sides 2a and 2b called the main rectangle of a hyperbola
. 3. It follows from equation (11.9) that the minuend is not less than one, i.e., that or . This means that the points of the hyperbola are located to the right of the line (the right branch of the hyperbola) and to the left of the line (the left branch of the hyperbola). 4. From the equation (11.9) of the hyperbola, it can be seen that when it increases, then it also increases. This follows from the fact that the difference keeps a constant value equal to one. It follows from what has been said that the hyperbola has the shape shown in Figure 54 (a curve consisting of two unbounded branches). Asymptotes of a hyperbola
The line L is called the asymptote Let us show that the hyperbola has two asymptotes: Since the lines (11.11) and the hyperbola (11.9) are symmetrical with respect to the coordinate axes, it suffices to consider only those points of the indicated lines that are located in the first quadrant. Take on a straight line a point N having the same abscissa x as a point on a hyperbola As you can see, as x increases, the denominator of the fraction increases; numerator is a constant value. Therefore, the length of the segment When constructing a hyperbola (11.9), it is advisable to first construct the main rectangle of the hyperbola (see Fig. 57), draw lines passing through the opposite vertices of this rectangle - the asymptotes of the hyperbola and mark the vertices and , hyperbola. The equation of an equilateral hyperbola. whose asymptotes are the coordinate axes The asymptotes of an equilateral hyperbola have equations and are therefore bisectors of the coordinate angles. Consider the equation of this hyperbola in a new coordinate system (see Fig. 58), obtained from the old one by rotating the coordinate axes by an angle. We use the formulas for the rotation of the coordinate axes: We substitute the values of x and y in equation (11.12): The equation of an equilateral hyperbola, for which the axes Ox and Oy are asymptotes, will have the form . More about hyperbole eccentricity
hyperbola (11.9) is the ratio of the distance between the foci to the value of the real axis of the hyperbola, denoted by ε: Since for a hyperbola , the eccentricity of the hyperbola is greater than one: . Eccentricity characterizes the shape of a hyperbola. Indeed, it follows from equality (11.10) that i.e. and This shows that the smaller the eccentricity of the hyperbola, the smaller the ratio - of its semi-axes, which means that the more its main rectangle is extended. The eccentricity of an equilateral hyperbola is . Really, Focal radii Straight lines are called directrixes of a hyperbola. Since for the hyperbola ε > 1, then . This means that the right directrix is located between the center and the right vertex of the hyperbola, the left directrix is between the center and the left vertex. The directrixes of a hyperbola have the same property as the directrixes of an ellipse. The curve defined by the equation is also a hyperbola, the real axis 2b of which is located on the Oy axis, and the imaginary axis 2 a- on the Ox axis. In Figure 59, it is shown as a dotted line. Obviously, the hyperbolas and have common asymptotes. Such hyperbolas are called conjugate. 11.5. Parabola Canonical parabola equation A parabola is the set of all points in a plane, each of which is equally distant from a given point, called the focus, and a given line, called the directrix. The distance from the focus F to the directrix is called the parameter of the parabola and is denoted by p (p > 0). 1. In equation (11.13), the variable y is included in an even degree, which means that the parabola is symmetric about the Ox axis; the x-axis is the axis of symmetry of the parabola. 2. Since ρ > 0, it follows from (11.13) that . Therefore, the parabola is located to the right of the y-axis. 3. When we have y \u003d 0. Therefore, the parabola passes through the origin. 4. With an unlimited increase in x, the module y also increases indefinitely. The parabola has the form (shape) shown in Figure 61. The point O (0; 0) is called the vertex of the parabola, the segment FM \u003d r is called the focal radius of the point M. Equations , , ( p>0) also define parabolas, they are shown in Figure 62 It is easy to show that the graph of a square trinomial, where , B and C are any real numbers, is a parabola in the sense of its definition above. 11.6. General equation of second order lines Equations of curves of the second order with axes of symmetry parallel to the coordinate axes Let us first find the equation of an ellipse centered at a point whose symmetry axes are parallel to the coordinate axes Ox and Oy and the semiaxes are respectively equal to a and b. Let us place in the center of the ellipse O 1 the origin of the new coordinate system , whose axes and semi-axes a and b(see fig. 64): And finally, the parabolas shown in Figure 65 have corresponding equations. The equation The equations of an ellipse, hyperbola, parabola and the equation of a circle after transformations (open brackets, move all terms of the equation in one direction, bring like terms, introduce new notation for the coefficients) can be written using a single equation of the form where the coefficients A and C are not equal to zero at the same time. The question arises: does any equation of the form (11.14) determine one of the curves (circle, ellipse, hyperbola, parabola) of the second order? The answer is given by the following theorem. Theorem 11.2. Equation (11.14) always defines: either a circle (for A = C), or an ellipse (for A C > 0), or a hyperbola (for A C< 0), либо
параболу (при А×С= 0). При этом возможны случаи вырождения: для эллипса (окружности)
- в точку или мнимый эллипс (окружность), для гиперболы - в пару пересекающихся
прямых, для параболы - в пару параллельных прямых. General equation of the second order Consider now the general equation of the second degree with two unknowns: It differs from equation (11.14) by the presence of a term with the product of coordinates (B¹ 0). It is possible, by rotating the coordinate axes by an angle a, to transform this equation so that the term with the product of coordinates is absent in it. Using formulas for turning axes Let's express the old coordinates in terms of the new ones: We choose the angle a so that the coefficient at x "y" vanishes, i.e., so that the equality Thus, when the axes are rotated through an angle a that satisfies condition (11.17), equation (11.15) reduces to equation (11.14). Conclusion: the general equation of the second order (11.15) defines on the plane (except for the cases of degeneracy and decay) the following curves: circle, ellipse, hyperbola, parabola. Note: If A = C, then equation (11.17) loses its meaning. In this case cos2α = 0 (see (11.16)), then 2α = 90°, i.e. α = 45°. So, at A = C, the coordinate system should be rotated by 45 °.Parametric equation of an ellipse
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The simplest curve of the second order is a circle. Recall that a circle of radius R centered at a point is the set of all points Μ of the plane that satisfy the condition . Let a point in a rectangular coordinate system have coordinates x 0, y 0 a - an arbitrary point of the circle (see Fig. 48).
(11.2)
.
(11.4)
. Its center is at the point 
, and the radius
.
, then equation (11.3) has the form
.
. In this case, they say: “the circle has degenerated into a point” (has zero radius).
, then equation (11.4), and hence the equivalent equation (11.3), will not determine any line, since the right side of equation (11.4) is negative, and the left side is not negative (say: “imaginary circle”).
Denote the foci by F1 and F2, the distance between them in 2 c, and the sum of the distances from an arbitrary point of the ellipse to the foci - through 2 a(see fig. 49). By definition 2 a > 2c, i.e. a
> c.
(11.6)
(11.7)
2. Find the points of intersection of the ellipse with the coordinate axes. Putting , we find two points and , at which the axis intersects the ellipse (see Fig. 50). Putting in equation (11.7), we find the points of intersection of the ellipse with the axis: and . points A 1 ,
A2 , B1, B2 called the vertices of the ellipse. Segments A 1 A2 and B1 B2, as well as their lengths 2 a and 2 b are called respectively major and minor axes ellipse. Numbers a and b are called large and small, respectively. axle shafts ellipse.
Let M(x; y) be an arbitrary point of the ellipse with foci F 1 and F 2 (see Fig. 51). The lengths of the segments F 1 M=r 1 and F 2 M = r 2 are called the focal radii of the point M. Obviously,
Theorem 11.1. If is the distance from an arbitrary point of the ellipse to some focus, d is the distance from the same point to the directrix corresponding to this focus, then the ratio is a constant value equal to the eccentricity of the ellipse:
.
Denote the foci by F1 and F2 the distance between them through 2s, and the modulus of the difference in distances from each point of the hyperbola to the foci through 2a. A-priory 2a < 2s, i.e. a < c.
or , i.e. After simplifications, as was done when deriving the ellipse equation, we get
canonical equation of a hyperbola
(11.9)
(11.10)
of an unbounded curve K if the distance d from point M of curve K to this line tends to zero as the point M moves along the curve K indefinitely from the origin. Figure 55 illustrates the concept of an asymptote: the line L is an asymptote for the curve K.
(11.11)
(see Fig. 56), and find the difference ΜN between the ordinates of the straight line and the branch of the hyperbola:
ΜN tends to zero. Since ΜN is greater than the distance d from the point Μ to the line, then d even more so tends to zero. Thus, the lines are asymptotes of the hyperbola (11.9).
Hyperbola (11.9) is called equilateral if its semiaxes are equal (). Its canonical equation
(11.12)
.
and 
for the points of the right branch of the hyperbola have the form and , and for the left - 
and 
.
To derive the parabola equation, we choose the Oxy coordinate system so that the Oxy axis passes through the focus F perpendicular to the directrix in the direction from the directrix to F, and the origin O is located in the middle between the focus and the directrix (see Fig. 60). In the selected system, the focus F has coordinates , and the directrix equation has the form , or .
